Take a look at what the Lyman manual has to say about the 444, and compare their velocities to what some other ballistics programs report. What you will find is that the 444 velocity is dependent upon barrel length. Most ballistics programs use a 24 inch barrel for testing and report MV of around 2300fps for a 240gr bullet. Lyman's testing for that cartridge uses an 18 inch ported barrel and reports velocity for the same bullet of around 1600fps. I shoot the 444 through an 18 inch ported Winchester Timber Carbine and I find that the Lyman manual is pretty close. I've never tried to shoot mine past 100 yards but isnipe indicates a 37MOA drop at 400 yards with velocity of around 900fps and energy less than 450 foot pounds. Not much of a deer gun at those distances. As a result, I pulled the scope off and shoot iron sights for close range hogs and such. It hits hogs super hard inside of 100 yards.
Regarding handloading, I've only tried Varget for powder which is clearly not fast enough for complete burn inside the 18 inch barrel. You can tell because of all the unburnt powder on the bench after firing. So, with the triple 4, a longer barrel is better for velocity but you give up a lot in weight and convenience which is why I bought mine. I love the carbine length for carrying around.
The 444 built a reputation as a hard hitting short range brush gun. With short barrel velocity, 300-400 yards would be the ragged edge for the cartridge, maybe beyond the edge. You decide. Using a 22 inch barrel should give you most of that velocity back so make sure you use a chrono to verify your MV. As an aside, I've never found accuracy to be that great so that's another mark against the 444 for long range. And finally, the little Winchester kicks like a mule even with its ported barrel.
So, let us know how it goes, especially if you find a load that shoots accurately with decent velocity.